What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$ with an error term $E(T) = O(T^{\alpha+\epsilon})$ with $\alpha=1/2$ or $\alpha=1/3$?

Note I am not asking for the best $\alpha$; I am asking for different ways - the simpler the better - with the intention of choosing one of them to make explicit.

Methods I know:

Proofs based on Atkinson's 1949 formula. This formula yields $\alpha=1/2$ automatically and $\alpha=1/3$ after some smoothing and some work. The problem is that there is an asymptotically tiny error term that seems hard to make explicit without applying the saddle-point method explicitly (twice) - something that I know to be time-consuming and sometimes painful.

Balasubramanian (1978). Again, this doesn't look simple.

Ingham/Atkinson (1939). Short and relatively simple. Yields $E(T) = O(T^{1/2} \log^2 T)$.

I'm also curious about the same question (in the same way) for $$\int_0^T \left|\zeta\left(\sigma + i t\right)\right|^2 dt,$$ where $0\leq \sigma <1$.